Circular and Periodic Functions

The Unit Circle

Radians

A radian is the size of the angle subtended by an arc the same length as the radius of the circle.

Since the circumference of a circle is 2πr, there are 2π radians in 360°.

One radian is equal to ${360°}/{2π} = 57.2957795°$. Since π is irrational, the radian cannot be expressed exactly in degrees.

Arcs

The length of an arc = $({θ}/{2π})(2πr) = rθ$

This formula assumes the angle is given in radians.

Sectors

The area of a sector = $({θ}/{2π})(πr^2)={θr^2}/2$

where r the radius of the circle, and the sector subtends the central angle θ.

Graphing Circular Functions

Solving equations with the unit circle

If $-2π ≤ x ≤ 2π$, there are a number of solutions to an equation such as: $sinx=1/{√2}$

The first quadrant solution is $π/4$ (45°). However, the sine of $π - π/4 = {3π}/4$, in the second quadrant, is also $1/{√2}$. But also sin${-5π}/4$ and sin${-7π}/4$ give solutions of $1/{√2}$.

If the domain is not limited to one cycle ($-2π ≤ x ≤ 2π$), then sin${9π}/4$, sin${11π}/4$, sin${17π}/4$, sin${19π}/4$, etc. are also solutions. And the periodic function could be extended in the negative direction as well: sin${-13π}/4$, sin${-15π}/4$, etc.

Modelling with Sine and Cosine

A system with periodic motion can be described by an equation. If the motion is simple harmonic or rotational, the equation can be a sinusoidal function of time, t, the starting position, h(0), and a periodic factor. An example is a Ferris Wheel:

$$H(t)=rsin({2π}/T(t-T/4))+(H(0) + r)$$

where T is the period of one rotation, H(0) is the starting height, and r the radius.

Ferris Wheel Example

$h(t)=60cos({2π}/{30}(t-15))+ 60$

At time $t=0$, the equation reduces to $h(t)=60cos(-π)+ 60 $

$= -60 + 60 = 0$: the starting position is 0.

At time $t=15$, the equation reduces to $h(t)=60cos(0)+ 60 $

$= 60 + 60 = 120$: the height at $t=15$ seconds is 120m. Since the maximum value of cos(x) is 1, this is the maximum height reached.

At time $t=30$, the equation reduces to $h(t)=60cos(π)+ 60 $

$= -60 + 60 = 0$: the height at $t=30$ seconds is once again 0m. The motion is periodic with a period of one cycle of 30 seconds.

General Periodic Motion

Since a sine or cosine can take a value of -1, the zero point is established by $M$ = maximum height.

The angular speed of the motion is described by the argument of the cosine or sine: in our example $({2π}/{30}(t-15))$. In other words, a full cycle (2π radians) is made every $p$ seconds ($p$ = period).